Lyapunov Function for Two-species Mutualism Model with Constant Harvesting

  • Rusliza Ahmad


In this paper, the researcher proposes a simple mathematical model consisting of mutualistic interactions among two-species with constant harvesting. Mutualism is one kind of interaction that ends up being a win-win situation for both species involved. The interacting species benefit from this interaction and ultimately are better adapted for continuous existence. The harvesting function is implemented to describe the rate of removal of the species. This paper aims to investigate the global stability of the unique positive equilibrium point of the model. The global stability of the model is studied by using Lyapunov function method. By constructing a suitable Lyapunov function, it has been proven that the unique positive equilibrium point is globally asymptotically stable in a nonlinear system. Finally, numerical simulation is shown to illustrate theoretical results and to simulate the trajectories around the stable equilibrium point. From the numerical analysis, it is observed that both the species persist.


[1] L. L. Rockwood, Introduction to Population Ecology. Hoboken, USA: John Wiley and Sons, 2015.
[2] D. H. Boucher, S. James and K. H. Keeler, “The Ecology of Mutualism,” Annual Review of Ecology and Systematic, vol .13, pp. 315 – 347, 1982.
[3] R. Ahmad, “Global Stability of Two-Species Mutualism Model with Proportional Harvesting,” International Journal of Advanced and Applied Sciences, vol. 4, no. 7, pp. 74 – 79, 2017.
[4] S. Raghukumar, Fungi in Coastal and Oceanic Marine Ecosystems: Marine Fungi. India:Springer, 2017, pp.120 – 121.
[5] C. Starr, R. Taggart, C. Evers and L. Starr, Biology: The Unity and Diversity of Life, 13th ed. Boston: Cengage Learning, 2012.
[6] R. Benz, Ecology and Evolution: Islands of Change. Virginia, USA: NSTA Press, 2000.
[7] D. H. Janzen, The Natural History of Mutualism. In: D. L. Boucher. Editors. The Biology of Mutualism: Ecology and Evolution. Oxford University Press, New York, USA, 1985, pp.40 –99.
[8] P. J. Morin, Community Ecology. Hoboken, USA: John Wiley and Sons, 2011.
[9] K. S. Cheng, S. B. Hsu and S. S. Lin, “Some Results on Global Stability of a Predator-Prey System,” Journal of Mathematical Biology, vol. 12, no. 1, pp. 115 – 126, 1981.
[10] B. S. Goh, “Stability in Models of Mutualism,” The American Naturist, vol. 113, no. 2, pp. 261– 275, 1979.
[11] B. R. Reddy, K. L. Narayan, and N. C. Pattabhiramacharyulu, “On Global Stability of Two Mutually Interacting Species with Limited Resources for Both the Species,” International Journal of Contemporary Mathematical Sciences, vol. 6, no. 9, pp. 401 – 407, 2011.
[12] P. Georgescu, H. Zhang and D. Maxin, “The Global Stability of Coexisting Equilibria for Three Models of Mutualism,” Mathematical Biosciences and Engineering, vol. 13, no. 1, pp. 101 – 118, 2016.
[13] K. Yang, X. Xie and F. Chen, “Global Stability of a Discrete Mutualism Model,” Abstract and Applied Analysis, vol. 2014, Article ID 928726, 6 pages, 2014.
[14] C. Lei, “Dynamic Behaviors of a Stage-Structured Commensalism System,” Advances in Difference Equations, vol.301, no.2018, 2018.
[15] R. Ouncharoen, S. Pinjai, T. Dumrongpokaphan and Y. Lenbury, “Global Stability Analysis of Predator-Prey Model with Harvesting and Delay,” Thai Journal of Mathematics, vol. 8, no. 3, pp. 589 – 605, 2012.
[16] A. Martin, Predator-Prey Models with Delays and Prey Harvesting, Unpublished Master’s Thesis. Dalhousie University Halifix, Nova Scotia, 1999. [17] C. V. D. León, “Lyapunov Function for Two-Species Cooperative Systems,” Applied Mathematics and Computation, vol. 219, no. 5, pp. 2493 – 2497, 2012.
[18] N. Supajaidee and S. Moonchai, “Stability Analysis of a Fractional-Order-Two-Species Facultative Mutualism Model with Harvesting,” Advances in Difference Equations, vol. 372, no. 2017, 2017.
[19] K. D. Do and J. Pan, Control of Ships and Underwater Vehicles. Berlin, Germany: Springer Science and Business Media, 2009.
[20] W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems. Hoboken, USA: John Wiley and Sons, 1992.
[21] T. H. Fay and J. C. Greeff, “Lion, Wildebeest and Zebra: A Predator-Prey Model,” Ecological Modeling, vol. 196, no. 1, pp. 237 – 244, 2006.
How to Cite
AHMAD, Rusliza. Lyapunov Function for Two-species Mutualism Model with Constant Harvesting. Mathematical Sciences and Informatics Journal, [S.l.], v. 1, n. 2, p. 1-11, nov. 2020. ISSN 2735-0703. Available at: <>. Date accessed: 07 dec. 2022. doi:

Most read articles by the same author(s)

Obs.: This plugin requires at least one statistics/report plugin to be enabled. If your statistics plugins provide more than one metric then please also select a main metric on the admin's site settings page and/or on the journal manager's settings pages.